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Full text of "Treatise On Analysis Vol-Ii"

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The notion of a limit is defined as in (3.13) for arbitrary topological spaces,
and the criterion (3.13.1) is merely another formulation of the definition.
However, the conclusion of (3.13.3) is no longer necessarily true: for example,
if E is a set endowed with the chaotic topology (12.1.1), every sequence of
points of E has every point of E as a limit.

A topological space E is said to be Hausdorff, and its topology is said to be
a Hausdorff topology, if it satisfies the following " Hausdorff axiom":

Given any two distinct points a, b in E, there exists a neighborhood U of a and a
neighborhood Vofb which do not intersect.

Every metrizable space is Hausdorff.

(12.3.1)    Let E be a topological space, A a subset of E, and let a be a cluster
point of A. Then a mapping f of A into a Hausdorff space E' has at most one
limit at the point a with respect to A.

For if a'9 V were two distinct limits of/at the point a, there would exist
neighborhoods U', V of d, V, respectively, having no point in common. But
by hypothesis there would be a neighborhood W of a in E such that
/(W n A) c U' and/(W n A) c V, and this is absurd because W n A ^ 0.

Hence we may continue to use the notation     lim   f(x) to denote the

jceA, x -* a

unique limit of/at the point a. The propositions of (3.13) which do not refer
to distances will remain valid for mappings (in particular for sequences) from
an arbitrary topological space to a Hausdorff space, with the exception of
(3.13.13) and (3.13.14).

(12.3.2)    Every subspace of a Hausdorff space is Hausdorff.

(12.3.3)    Every topology which is finer than a Hausdorff topology is Hausdorff.
These are immediate consequences of the definitions.

(12.3.4)    In a Hausdorff space E, every finite subset is closed.

For if (flj)i^rt is a finite sequence of points in E, a point b distinct from
all the at cannot be a cluster point of the set of the at, because for each i thereinition, the condition (b)